In mathematics, mimesis is the quality of a numerical method which imitates some properties of the continuum problem. The goal of
numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...
is to approximate the continuum, so instead of solving a
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
one aims to solve a discrete version of the continuum problem. Properties of the continuum problem commonly imitated by numerical methods are
conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, ...
s,
solution symmetries, and fundamental identities and theorems of vector and tensor calculus like the
divergence theorem.
[.]
Both
finite difference
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for t ...
or
finite element method
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
can be mimetic; it depends on the properties that the method has.
For example, a mixed finite element method applied to
Darcy flows strictly
conserves the mass of the flowing fluid.
The term ''
geometric integration'' denotes the same philosophy.
References
{{Mathapplied-stub
Numerical differential equations